11 research outputs found
Entanglement entropy in collective models
We discuss the behavior of the entanglement entropy of the ground state in
various collective systems. Results for general quadratic two-mode boson models
are given, yielding the relation between quantum phase transitions of the
system (signaled by a divergence of the entanglement entropy) and the
excitation energies. Such systems naturally arise when expanding collective
spin Hamiltonians at leading order via the Holstein-Primakoff mapping. In a
second step, we analyze several such models (the Dicke model, the two-level BCS
model, the Lieb-Mattis model and the Lipkin-Meshkov-Glick model) and
investigate the properties of the entanglement entropy in the whole parameter
range. We show that when the system contains gapless excitations the
entanglement entropy of the ground state diverges with increasing system size.
We derive and classify the scaling behaviors that can be met.Comment: 11 pages, 7 figure
Two-spin entanglement distribution near factorized states
We study the two-spin entanglement distribution along the infinite
chain described by the XY model in a transverse field; closed analytical
expressions are derived for the one-tangle and the concurrences ,
being the distance between the two possibly entangled spins, for values of the
Hamiltonian parameters close to those corresponding to factorized ground
states. The total amount of entanglement, the fraction of such entanglement
which is stored in pairwise entanglement, and the way such fraction distributes
along the chain is discussed, with attention focused on the dependence on the
anisotropy of the exchange interaction. Near factorization a characteristic
length-scale naturally emerges in the system, which is specifically related
with entanglement properties and diverges at the critical point of the fully
isotropic model. In general, we find that anisotropy rule a complex behavior of
the entanglement properties, which results in the fact that more isotropic
models, despite being characterized by a larger amount of total entanglement,
present a smaller fraction of pairwise entanglement: the latter, in turn, is
more evenly distributed along the chain, to the extent that, in the fully
isotropic model at the critical field, the concurrences do not depend on .Comment: 14 pages, 6 figures. Final versio
Renyi Entropy of the XY Spin Chain
We consider the one-dimensional XY quantum spin chain in a transverse
magnetic field. We are interested in the Renyi entropy of a block of L
neighboring spins at zero temperature on an infinite lattice. The Renyi entropy
is essentially the trace of some power of the density matrix of the
block. We calculate the asymptotic for analytically in terms of
Klein's elliptic - function. We study the limiting entropy as a
function of its parameter . We show that up to the trivial addition
terms and multiplicative factors, and after a proper re-scaling, the Renyi
entropy is an automorphic function with respect to a certain subgroup of the
modular group; moreover, the subgroup depends on whether the magnetic field is
above or below its critical value. Using this fact, we derive the
transformation properties of the Renyi entropy under the map and show that the entropy becomes an elementary function of the
magnetic field and the anisotropy when is a integer power of 2, this
includes the purity . We also analyze the behavior of the entropy as
and and at the critical magnetic field and in the
isotropic limit [XX model].Comment: 28 Pages, 1 Figur
On the entanglement entropy for a XY spin chain
The entanglement entropy for the ground state of a XY spin chain is related
to the corner transfer matrices of the triangular Ising model and expressed in
closed form.Comment: 4 pages, 2 figure
Asymptotics of Toeplitz Determinants and the Emptiness Formation Probability for the XY Spin Chain
We study an asymptotic behavior of a special correlator known as the
Emptiness Formation Probability (EFP) for the one-dimensional anisotropic XY
spin-1/2 chain in a transverse magnetic field. This correlator is essentially
the probability of formation of a ferromagnetic string of length in the
antiferromagnetic ground state of the chain and plays an important role in the
theory of integrable models. For the XY Spin Chain, the correlator can be
expressed as the determinant of a Toeplitz matrix and its asymptotical
behaviors for throughout the phase diagram are obtained using
known theorems and conjectures on Toeplitz determinants. We find that the decay
is exponential everywhere in the phase diagram of the XY model except on the
critical lines, i.e. where the spectrum is gapless. In these cases, a power-law
prefactor with a universal exponent arises in addition to an exponential or
Gaussian decay. The latter Gaussian behavior holds on the critical line
corresponding to the isotropic XY model, while at the critical value of the
magnetic field the EFP decays exponentially. At small anisotropy one has a
crossover from the Gaussian to the exponential behavior. We study this
crossover using the bosonization approach.Comment: 40 pages, 9 figures, 1 table. The poor quality of some figures is due
to arxiv space limitations. If You would like to see the pdf with good
quality figures, please contact Fabio Franchini at
"[email protected]
Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion
We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, \texttt{arXiv:1104.1004}, for the reduced density of states of two disjoint intervals with lattice sites , which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher-Hartwig symbol using the the Riemann--Hilbert method
Entanglement and Density Matrix of a Block of Spins in AKLT Model
We study a 1-dimensional AKLT spin chain, consisting of spins in the bulk
and at both ends. The unique ground state of this AKLT model is described
by the Valence-Bond-Solid (VBS) state. We investigate the density matrix of a
contiguous block of bulk spins in this ground state. It is shown that the
density matrix is a projector onto a subspace of dimension . This
subspace is described by non-zero eigenvalues and corresponding eigenvectors of
the density matrix. We prove that for large block the von Neumann entropy
coincides with Renyi entropy and is equal to .Comment: Revised version, typos corrected, references added, 31 page